Bieri, Geoghegan and Kochloukova computed the BNSR-invariants Σ^m(F) of Thompson’s group F for all m. We recompute these using entirely geometric techniques, making use of the Stein–Farley CAT(0) cube complex X on which F acts.

We explore the ideal structure of the reduced C∗-algebra of R. Thompson’s group T. We show that even though T has trace, one cannot use the Kesten Condition to verify that the reduced C∗-algebra of T is simple. At the time of the initial writing of this chapter, there had been no example group for which it was known that the Kesten Condition would fail to prove simplicity, even though the group has trace. Motivated by this first result, we describe a class of groups where even if the group has trace, one cannot apply the Kesten Condition to verify the simplicity of those groups' reduced C∗-algebras. We also offer an apparently weaker condition to test for the simplicity of a group's reduced C∗-algebra, and we show this new test is still insufficient to show that the reduced C∗-algebra of T is simple. Separately, we find a controlled version of a Ping-Pong Lemma which allows one to find non-abelian free subgroups in groups of homeomorphisms of the circle generated by elements with rational rotation number. We use our Ping-Pong Lemma to find a simple converse to a theorem of Uffe Haagerup and Kristian Knudsen Olesen.

Let G be a group and H be a subgroup of G. We say that H is left relatively convex in G if the left G-set G/H has at least one G-invariant order; when G is left orderable, this holds if and only if H is convex in G under some left ordering of G. We give a criterion for H to be left relatively convex in G that generalizes a famous theorem of Burns and Hale and has essentially the same proof. We show that all maximal cyclic subgroups are left relatively convex in free groups, in right-angled Artin groups, and in surface groups that are not the Klein-bottle group. The free-group case extends a result of Duncan and Howie. More generally, every maximal m-generated subgroup in a free group is left relatively convex. The same result is valid, with some exceptions, for compact surface groups. Maximal m-generated abelian subgroups in right-angled Artin groups are left relatively convex. If G is left orderable, then each free factor of G is left relatively convex in G. More generally, for any graph of groups, if each edge group is left relatively convex in each of its vertex groups, then each vertex group is left relatively convex in the fundamental group; this generalizes a result of Chiswell.

We give a simple technique to compute the distance between two points in an n-dimensional Euclidean simplex, where the points are given in barycentric coordinates, using only the edge lengths of that simplex. We then use this technique to verify a few computations which will be used in subsequent papers. The most important application is a formula for intrinsically computing the volume of a Euclidean simplex, which is more efficient (and more natural) than any previously documented methods.

We fully describe the horofunction boundary δhL2 with the word metric associated with the generating set {t, at} (i.e. the metric arising in the Diestel–Leader graph DL(2, 2)). The visual boundary δ∞L2 with this metric is a subset of δhL2. Although δ∞L2 does not embed continuously in δhL2, it naturally splits into two subspaces, each of which is a punctured Cantor set and does embed continuously. The height function on DL(2, 2) provides a natural stratification of δhL2, in which countably many non-Busemann points interpolate between the two halves of δ∞L2. Furthermore, the height function and its negation are themselves non-Busemann horofunctions in δhL2 and are global fixed points of the action of L2.

Let G be a group, and let S be a finite subset of G that generates G as a monoid. The co-word problem is the collection of words in the free monoid S∗that represent non-trivial elements of G. A current conjecture, based originally on a conjecture of Lehnert and modified into its current form by Bleak, Matucci, and Neunhöffer, says that Thompson’s group V is a universal group with context-free co-word problem. It is thus conjectured that a group has a context-free co-word problem exactly if it is a finitely generated subgroup of V. Hughes introduced the class FSS of groups that are determined by finite similarity structures. An FSS group acts by local similarities on a compact ultrametric space. Thompson’s group V is a representative example, but there are many others.We show that FSS groups have context-free co-word problem under a minimal additional hypothesis. As a result, we can specify a subfamily of FSS groups that are potential counterexamples to the conjecture.

We discuss the connection between Chevalley’s definition of a covering space and the usual definition given in an introductory topology course. Then we indicate how some theorems about the covering groups of a topological group can be proved from the global point of view, without using local isomorphisms between topological groups.

The aim of this chapter is to provide some new tools to aid the study of decomposition complexity, a notion introduced by Guentner, Tessera and Yu. In this chapter, three equivalent definitions for decomposition complexity are established. We prove that metric spaces with finite hyperbolic dimension have finite (weak) decomposition complexity, and we prove that the collection of metric families that are coarsely embeddable into Hilbert space is closed under decomposition. A method for showing that certain metric spaces do not have finite decomposition complexity is also discussed.

This is a summary, written by the first-named author, of his joint work with Ross Geoghegan over the past years. Most of the material is available in detail in the preprint “Limit sets for modules over groups on cat(0) spaces – from the Euclidean to the hyperbolic,” available at http://arxiv.org/abs/1306.3403, and I will occasionally refer to specific detail in that paper. Other parts of our joint work - results mostly concerned with extending concepts and results from that paper to higher dimensions – will also be mentioned but are still in preparation.

An important “stability” theorem in shape theory, due to D. A. Edwards and R. Geoghegan, characterizes those compacta having the same shape as a finite CW complex. In this chapter we present a straightforward and self-contained proof of that theorem.

Let G be a finitely generated group, and Σ a finite subset that generates G as a monoid. The word problem of G with respect to Σ consists of all words in the free monoid Σ* that are equal to the identity in G. The co-word problem of G with respect to Σ is the complement in Σ* of the word problem. We say that a group G is coCF if its co-word problem with respect to some (equivalently, any) finite generating set Σ is a context-free language. We describe a generalized Thompson group V(G,θ) for each finite group G and homomorphismθ: G → G. Our group is constructed using the cloning systems introduced by Witzel and Zaremsky. We prove that V(G,θ) is coCF for any homomorphism θ and finite group G by constructing a pushdown automaton and showing that the co-word problem of V(G,θ) is the cyclic shift of the language accepted by our automaton. Demonstrative subgroups of V, introduced by Bleak and Salazar-Diaz, can be used to construct embeddings of certain wreath products and amalgamated free products into V. We extend the class of known finitely generated demonstrative subgroups of V to include all virtually cyclic groups.